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## G = C4⋊C4⋊26D10order 320 = 26·5

### 9th semidirect product of C4⋊C4 and D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4⋊26D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C4⋊C4⋊26D10
 Lower central C5 — C2×C10 — C4⋊C4⋊26D10
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C4⋊C426D10
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1294 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], D4 [×14], Q8 [×2], C23, C23 [×10], D5 [×6], C10 [×3], C10 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×11], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×5], D10 [×4], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×2], C23×C4, C2×C4○D4, C4×D5 [×4], C4×D5 [×10], D20 [×10], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C22.19C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20, C2×D20 [×4], Q82D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C22×C20, Q8×C10, C23×D5, Dic54D4 [×2], C22⋊D20 [×2], C4⋊C47D5, D208C4 [×2], D10.13D4 [×2], C4⋊D20, C207D4, D103Q8, C5×C22⋊Q8, D5×C22×C4, C2×Q82D5, C4⋊C426D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.19C24, D4×D5 [×2], Q82D5 [×2], C23×D5, C2×D4×D5, C2×Q82D5, D5×C4○D4, C4⋊C426D10

Smallest permutation representation of C4⋊C426D10
On 80 points
Generators in S80
```(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 77 26 72)(22 78 27 73)(23 79 28 74)(24 80 29 75)(25 71 30 76)(31 65 36 70)(32 66 37 61)(33 67 38 62)(34 68 39 63)(35 69 40 64)
(1 33 6 27)(2 23 7 39)(3 35 8 29)(4 25 9 31)(5 37 10 21)(11 36 16 30)(12 26 17 32)(13 38 18 22)(14 28 19 34)(15 40 20 24)(41 65 46 71)(42 77 47 61)(43 67 48 73)(44 79 49 63)(45 69 50 75)(51 70 56 76)(52 72 57 66)(53 62 58 78)(54 74 59 68)(55 64 60 80)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)(41 44)(42 43)(45 50)(46 49)(47 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 80)(70 79)```

`G:=sub<Sym(80)| (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,77,26,72)(22,78,27,73)(23,79,28,74)(24,80,29,75)(25,71,30,76)(31,65,36,70)(32,66,37,61)(33,67,38,62)(34,68,39,63)(35,69,40,64), (1,33,6,27)(2,23,7,39)(3,35,8,29)(4,25,9,31)(5,37,10,21)(11,36,16,30)(12,26,17,32)(13,38,18,22)(14,28,19,34)(15,40,20,24)(41,65,46,71)(42,77,47,61)(43,67,48,73)(44,79,49,63)(45,69,50,75)(51,70,56,76)(52,72,57,66)(53,62,58,78)(54,74,59,68)(55,64,60,80), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,80)(70,79)>;`

`G:=Group( (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,77,26,72)(22,78,27,73)(23,79,28,74)(24,80,29,75)(25,71,30,76)(31,65,36,70)(32,66,37,61)(33,67,38,62)(34,68,39,63)(35,69,40,64), (1,33,6,27)(2,23,7,39)(3,35,8,29)(4,25,9,31)(5,37,10,21)(11,36,16,30)(12,26,17,32)(13,38,18,22)(14,28,19,34)(15,40,20,24)(41,65,46,71)(42,77,47,61)(43,67,48,73)(44,79,49,63)(45,69,50,75)(51,70,56,76)(52,72,57,66)(53,62,58,78)(54,74,59,68)(55,64,60,80), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,80)(70,79) );`

`G=PermutationGroup([(1,53,13,43),(2,54,14,44),(3,55,15,45),(4,56,16,46),(5,57,17,47),(6,58,18,48),(7,59,19,49),(8,60,20,50),(9,51,11,41),(10,52,12,42),(21,77,26,72),(22,78,27,73),(23,79,28,74),(24,80,29,75),(25,71,30,76),(31,65,36,70),(32,66,37,61),(33,67,38,62),(34,68,39,63),(35,69,40,64)], [(1,33,6,27),(2,23,7,39),(3,35,8,29),(4,25,9,31),(5,37,10,21),(11,36,16,30),(12,26,17,32),(13,38,18,22),(14,28,19,34),(15,40,20,24),(41,65,46,71),(42,77,47,61),(43,67,48,73),(44,79,49,63),(45,69,50,75),(51,70,56,76),(52,72,57,66),(53,62,58,78),(54,74,59,68),(55,64,60,80)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13),(15,20),(16,19),(17,18),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,40),(30,39),(41,44),(42,43),(45,50),(46,49),(47,48),(51,54),(52,53),(55,60),(56,59),(57,58),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,80),(70,79)])`

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 20 20 2 2 2 2 4 4 4 4 5 5 5 5 10 10 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 C4○D4 D10 D10 D10 D10 D4×D5 Q8⋊2D5 D5×C4○D4 kernel C4⋊C4⋊26D10 Dic5⋊4D4 C22⋊D20 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊D20 C20⋊7D4 D10⋊3Q8 C5×C22⋊Q8 D5×C22×C4 C2×Q8⋊2D5 C4×D5 C22⋊Q8 D10 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C4 C22 C2 # reps 1 2 2 1 2 2 1 1 1 1 1 1 4 2 4 4 4 6 2 2 4 4 4

Matrix representation of C4⋊C426D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 32 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 39 0 0 0 0 1 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 34 35 0 0 0 0 7 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1
,
 7 1 0 0 0 0 34 34 0 0 0 0 0 0 40 0 0 0 0 0 40 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,32,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;`

C4⋊C426D10 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes_{26}D_{10}`
`% in TeX`

`G:=Group("C4:C4:26D10");`
`// GroupNames label`

`G:=SmallGroup(320,1299);`
`// by ID`

`G=gap.SmallGroup(320,1299);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,794,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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